Tuesday, October 22, 2013

Where Is the Missing Dollar?

I was recently watching the Highway to Heaven episode "It's a Dog's Life."  Near the beginning of that episode, Jonathan gives Mark a mathematical riddle.  The riddle has shown up in other places than that Highway to Heaven episode.  As a matter of fact, my sixth grade teacher gave it to my class years ago.  I'll quote wikipedia's rendition of it:

"Three guests check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $25. To rectify this, he gives the bellhop $5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $1 and keep $2 for himself. Each guest got $1 back: so now each guest only paid $9; bringing the total paid to $27. The bellhop has $2. And $27 + $2 = $29 so, if the guests originally handed over $30, what happened to the remaining $1?"

I had to lay out thirty pennies to solve the riddle.  I wanted to make sure that the remaining one penny would not get lost!  What I found is that the riddle makes false assumptions.

For one, we're trying to account for all of the $30, right?  Why, then, does the riddle only consider the $27 that the guests paid to the hotel, plus the $2 that the bellhop kept for himself, while totally ignoring the $3 that the guests got as a refund (with $1 for each of the three guests)?  In short, why add the $27 and the $2, while ignoring the $3 that the bellhop let the guests have?  Actually, if we consider the $3, the riddle would not be what happened to the remaining $1, but rather where the two additional dollars came from, since $27+$2+$3=$32. 

That brings me to my second point: the riddle adds the $27 and the $2 together.  Actually, what one needs to do is subtract the $2 from the $27, not add them together.  When I was laying out the pennies, I set them out in three groups, representing the money that each guest gave to the hotel.  Each guest initially paid $10, then, after the refund, what each ended up paying to the hotel was $9, amounting to $27.  But I noticed a problem.  The guests were keeping $3, and the bellhop kept $2.  For the bellhop to get that $2, I needed to take $2 away from the $27.  There was no other way.  And remember that, in the riddle, the clerk gives the bellhop $5, which would leave the money that the hotel has at $25.  The hotel has $25, the guests have $3, and the bellhop has $2.  That's $30.  There is no missing dollar! 

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